Subgroups — C₂×#26018225 #36464211

Order 22784.

414 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151419	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018225	C₂	1	4	yes	maximal, direct product factor
4	C₂×#443×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	—	2	1	no	—
6	C₂×C₄×#10623494	—	2	1	no	—
7	#19151419	—	2	1	no	—
8	#19151419	C₂²	1	2	yes	complemented
9	C₈×#10623494	—	2	1	no	—
10	C₈×#10623494	C₂²	1	2	yes	complemented
11	C₂×#135×C₈₉	C₄	1	1	yes	characteristic
12	C₂×#131×C₈₉	—	2	1	no	—
13	#443×C₈₉	—	2	1	no	—
14	#443×C₈₉	—	2	1	no	—
15	#443×C₈₉	C₄	1	2	yes	—
16	#443×C₈₉	C₂²	1	2	yes	—
17	C₂×#108×C₈₉	C₄	1	1	yes	characteristic
18	C₂×#104×C₈₉	—	2	1	no	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×#10623494	—	2	2	no	—
21	#15667952	—	2	2	no	—
22	#15667952	—	4	1	no	—
23	C₂×#13083549	—	4	1	no	—
24	C₄×#10623494	—	2	2	no	—
25	C₄×#10623494	—	2	2	no	—
26	C₂×#41×C₈₉	—	2	1	no	—
27	#135×C₈₉	—	2	1	no	—
28	#135×C₈₉	—	2	1	no	—
29	#135×C₈₉	C₈	1	2	yes	complemented
30	#135×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₈₉	C₈	1	1	yes	characteristic, complemented
32	C₂×SD₁₆×C₈₉	—	2	1	no	—
33	C₂×D₈×C₈₉	C₈	1	1	yes	characteristic, complemented
34	#131×C₈₉	—	2	2	no	—
35	#131×C₈₉	—	2	2	no	—
36	#131×C₈₉	—	4	1	no	—
37	#131×C₈₉	—	4	1	no	—
38	C₂×M₁₆×C₈₉	—	2	1	no	—
39	C₂×M₁₆×C₈₉	—	4	1	no	—
40	C₂²×C₈×C₈₉	—	4	1	no	—
41	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	#108×C₈₉	—	2	1	no	—
45	#108×C₈₉	C₂×C₄	1	2	yes	—
46	#104×C₈₉	—	2	2	no	—
47	#104×C₈₉	—	2	2	no	—
48	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×#10623494	—	2	8	no	—
53	C₂×#10623494	—	2	4	no	—
54	#13083549	—	4	2	no	—
55	#41×C₈₉	—	2	2	no	—
56	#41×C₈₉	—	2	2	no	—
57	#41×C₈₉	—	4	1	no	—
58	#41×C₈₉	—	4	1	no	—
59	C₂×Q₈×C₈₉	—	2	1	no	—
60	C₂×D₄×C₈₉	—	2	1	no	—
61	C₂×D₄×C₈₉	—	4	1	no	—
62	C₂²×C₄×C₈₉	—	2	1	no	—
63	C₂²×C₄×C₈₉	—	4	1	no	—
64	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	Q₁₆×C₈₉	—	2	1	no	—
66	Q₁₆×C₈₉	C₂×C₈	1	2	yes	complemented
67	SD₁₆×C₈₉	—	2	2	no	—
68	SD₁₆×C₈₉	—	2	2	no	—
69	D₈×C₈₉	—	2	1	no	—
70	D₈×C₈₉	C₂×C₈	1	2	yes	complemented
71	M₁₆×C₈₉	—	2	2	no	—
72	M₁₆×C₈₉	—	4	1	no	—
73	M₁₆×C₈₉	—	4	2	no	—
74	M₁₆×C₈₉	—	4	2	no	—
75	C₂×C₈×C₈₉	—	4	1	no	—
76	C₂×C₈×C₈₉	—	4	2	no	—
77	C₂×C₈×C₈₉	—	4	2	no	—
78	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
81	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
82	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
83	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
84	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
85	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
86	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
87	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₈₉	—	2	1	no	—
91	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂³×C₈₉	—	4	1	no	—
94	C₂³×C₈₉	#104	1	1	yes	characteristic
95	Q₈×C₈₉	—	2	2	no	—
96	Q₈×C₈₉	—	4	1	no	—
97	D₄×C₈₉	—	2	2	no	—
98	D₄×C₈₉	—	4	2	no	—
99	D₄×C₈₉	—	4	2	no	—
100	D₄×C₈₉	—	4	1	no	—
101	C₂×C₄×C₈₉	—	2	4	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	4	1	no	—
105	C₂×C₄×C₈₉	—	4	2	no	—
106	C₂×C₄×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#104	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₈×C₈₉	—	4	2	no	—
114	C₈×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
115	C₈×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
116	C₈×C₈₉	#102	1	1	yes	characteristic
117	C₈×C₈₉	#102	1	1	yes	characteristic
118	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
119	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
120	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
122	#10623494	—	2	8	no	—
123	C₂²×C₈₉	—	2	2	no	—
124	C₂²×C₈₉	—	4	2	no	—
125	C₂²×C₈₉	—	8	2	no	—
126	C₂²×C₈₉	D₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₈₉	D₄⋊C₈	1	1	yes	characteristic
128	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
129	C₄×C₈₉	—	2	2	no	—
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	4	2	no	—
132	C₄×C₈₉	D₄⋊C₈	1	1	yes	characteristic
133	C₄×C₈₉	D₄⋊C₈	1	1	yes	characteristic
134	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
136	C₂×#1335	—	89	1	no	maximal, Sylow
137	C₂×C₈₉	—	2	1	no	—
138	C₂×C₈₉	—	2	1	no	—
139	C₂×C₈₉	—	8	2	no	—
140	C₂×C₈₉	#1335	1	2	yes	complemented
141	C₂×C₈₉	C₂×D₄⋊C₈	1	1	yes	characteristic
142	C₂×#443	D₈₉	1	1	yes	characteristic
143	C₂×#364	—	89	1	no	—
144	C₂×C₈²	—	89	1	no	—
145	#1335	—	89	4	no	—
146	C₈₉	C₂×#1335	1	1	yes	characteristic, complemented, Sylow
147	C₂×#135	Dic₈₉	1	1	yes	characteristic
148	C₂×#131	—	2	1	no	—
149	C₂×M₃₂	—	178	1	no	—
150	#443	—	2	1	no	—
151	#443	—	2	1	no	—
152	#443	Dic₈₉	1	2	yes	—
153	#443	C₂×D₈₉	1	2	yes	—
154	C₂×#108	Dic₈₉	1	1	yes	characteristic
155	C₂×#104	—	2	1	no	—
156	C₂×C₄×C₈	—	178	1	no	—
157	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
158	#364	—	89	2	no	—
159	#364	—	178	1	no	—
160	C₈²	—	89	2	no	—
161	C₈²	—	178	1	no	—
162	C₂×#41	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	—	2	1	no	—
165	#135	#10623494	1	2	yes	complemented
166	#135	C₂×Dic₈₉	1	2	yes	—
167	C₂×Q₁₆	#10623494	1	1	yes	characteristic, complemented
168	C₂×SD₁₆	—	2	1	no	—
169	C₂×D₈	#10623494	1	1	yes	characteristic, complemented
170	#131	—	2	2	no	—
171	#131	—	2	2	no	—
172	#131	—	4	1	no	—
173	#131	—	4	1	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	4	1	no	—
176	C₂²×C₈	—	4	1	no	—
177	C₂²×C₈	—	178	2	no	—
178	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
181	M₃₂	—	356	1	no	—
182	M₃₂	—	178	2	no	—
183	C₂×C₁₆	—	356	1	no	—
184	#108	—	2	1	no	—
185	#108	C₂×Dic₈₉	1	2	yes	—
186	#104	—	2	2	no	—
187	#104	—	2	2	no	—
188	C₄×C₈	—	178	2	no	—
189	C₄×C₈	—	178	2	no	—
190	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
191	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
192	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
204	Q₁₆	—	2	1	no	—
205	Q₁₆	C₂×#10623494	1	2	yes	complemented
206	SD₁₆	—	2	2	no	—
207	SD₁₆	—	2	2	no	—
208	D₈	—	2	1	no	—
209	D₈	C₂×#10623494	1	2	yes	complemented
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	178	4	no	—
218	C₂×C₈	—	178	8	no	—
219	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
221	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
222	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
223	C₂×C₈	#13083558	1	1	yes	characteristic
224	C₂×C₈	#13083558	1	1	yes	characteristic
225	C₂×C₈	#13083559	1	1	yes	characteristic
226	C₂×C₈	#13083559	1	1	yes	characteristic
227	C₂×C₈	#13083562	1	1	yes	characteristic
228	C₂×C₈	#13083562	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
231	C₄²	—	2	1	no	—
232	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
234	C₁₆	—	356	2	no	—
235	C₂³	—	4	1	no	—
236	C₂³	#15667939	1	1	yes	characteristic
237	Q₈	—	2	2	no	—
238	Q₈	—	4	1	no	—
239	D₄	—	2	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	1	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
251	C₂×C₄	#15667933	1	1	yes	characteristic
252	C₂×C₄	#15667933	1	1	yes	characteristic
253	C₂×C₄	#15667939	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
255	C₈	—	4	2	no	—
256	C₈	—	178	8	no	—
257	C₈	C₂²⋊#10623494	1	1	yes	characteristic
258	C₈	C₂²⋊#10623494	1	1	yes	characteristic
259	C₈	#15667933	1	1	yes	characteristic
260	C₈	#15667933	1	1	yes	characteristic
261	C₈	C₂²×#10623494	1	1	yes	characteristic
262	C₈	C₂×#13083558	1	1	yes	characteristic
263	C₈	C₂×#13083559	1	1	yes	characteristic
264	C₈	C₂×#13083562	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	D₄⋊#10623494	1	1	yes	characteristic
269	C₂²	D₄⋊#10623494	1	1	yes	characteristic
270	C₂²	C₂×#15667939	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	D₄⋊#10623494	1	1	yes	characteristic
275	C₄	D₄⋊#10623494	1	1	yes	characteristic
276	C₄	C₂×C₂²⋊#10623494	1	1	yes	characteristic
277	C₄	C₂×#15667933	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	8	2	no	—
281	C₂	#26018225	1	2	yes	direct product factor
282	C₂	C₂×D₄⋊#10623494	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#13083559
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×#443×C₈₉	C₂
socle	C₂²×C₈₉	D₄⋊C₈
prime radical	C₂×#135×C₈₉	C₄
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×#443	D₈₉
hypocenter	C₈₉	C₂×#1335
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#19151419	C₂	semidirect
2. C₂×C₈×#10623494	C₂	semidirect
3. #26018225	C₂	direct (N × H)
8. #19151419	C₂²	semidirect
10. C₈×#10623494	C₂²	semidirect
29. #135×C₈₉	C₈	semidirect
31. C₂×Q₁₆×C₈₉	C₈	semidirect
33. C₂×D₈×C₈₉	C₈	semidirect
66. Q₁₆×C₈₉	C₂×C₈	semidirect
70. D₈×C₈₉	C₂×C₈	semidirect
140. C₂×C₈₉	#1335	semidirect
146. C₈₉	C₂×#1335	semidirect
165. #135	#10623494	semidirect
167. C₂×Q₁₆	#10623494	semidirect
169. C₂×D₈	#10623494	semidirect
205. Q₁₆	C₂×#10623494	semidirect
209. D₈	C₂×#10623494	semidirect
281. C₂	#26018225	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19151419	2
C₂×C₈×#10623494	2
#26018225	2
C₂×#443×C₈₉	2
C₂×#1335	89

Subgroups — C2×#26018225 #36464211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018225 #36464211